Lesson 6 Sand Castles Solidify Understanding

Jump Start

Using the given information, draw and label a diagram of each of the following 3-D figures. Then find the volume of each solid.

1.

A right prism with a square base, on a side, and a height of .

2.

A right cylinder with a circular base of radius and a height of .

3.

A right cone with a circular base of radius and a height of .

Learning Focus

Use similarity to scale area and volume.

We have learned about dilation transformations in two dimensions. What happens when a solid undergoes a 3-D dilation?

How does such a dilation affect the area and volume of the solid?

Open Up the Math: Launch, Explore, Discuss

Benji, Chau, and Kassandra plan to enter a sand-castle-building contest sponsored by a local radio station. The winning team gets a private beach party at a local resort for all of their friends. To be selected for the competition, the team has to submit a drawing of their castle and verification that the design fits within the rules.

The three friends plan to build three similar castles, with each subsequent castle twice as big as the previous one. They hope that replicating the same design three times—while paying attention to the tiniest little details—will impress the judges with their creativity and sand-sculpting skill.

Benji is puzzling over a couple of questions on the application. They sound like math problems, and he wants Chau and Kassandra to make sure that he answers them correctly.

Please provide the following information about your sand sculpture:

  • What is the total area of the footprint of your planned sand sculpture?

This information will allow the planning committee to locate sand sculptures so the viewing public will have easy access to all sculptures. Remember that the total area occupied by your sculpture cannot exceed

  • What is the total volume of sand required to build your sand sculpture?

We will provide clean, sifted sand for each team, so we will not be liable for any debris or harmful substances that can be present in beach sand.

I certify that the above information is correct.

Signature of team leader: Date:

The friends have only designed one of the castles, since the others will be scaled-up versions of this one, with each castle “twice as big” as the previous one.

After studying one shape in the diagram, Benji said, “I calculated the area of the footprint of this shape for the smallest castle to be , so the next one will occupy , and the largest will occupy . That’s a total of , well within the limits.”

1.

What do you think of Benji’s comment? On graph paper, design a couple of possible “footprints” for a sand castle that will occupy of area. Then scale each design up so it is “twice as big,” and calculate the area. What do you notice?

2.

Imagine stacking cubes on your sand castle “footprints” to create a simple 3-D sculpture. Then scale up each design so it is “twice as big,” and calculate the volume. What do you notice?

3.

How did you interpret the phrase “twice as big” in your work on problems 1 and 2? Is your interpretation the same as Benji’s?

4.

To avoid confusion, it would be more appropriate for Benji and his friends to say they are going to “scale up” their initial sand castle by a factor of . If the “footprint” of a sand castle occupies , is it possible to calculate the area occupied by a sand castle that has been enlarged by a scale factor of , or is the area of the enlarged shape dependent upon the shape of the original figure? That is, do triangles, parallelograms, pentagons, etc., all scale up in the same way? Write a convincing argument explaining why or why not.

5.

What happens to the perimeter of the “footprint” of your sand castle when it is scaled up by a factor of ?

6.

Suppose your sand castle “footprint” was cut out of a piece of Styrofoam that is one-inch thick. What happens to the volume when this “3-D footprint” is scaled up by a factor of ?

Pause and Reflect

7.

The plans for the smallest sand castle include a rectangular prism that is high and has a square base with a side length of .

a.

What is the volume of sand required to make this prism in the smallest sand castle?

b.

What is the volume of sand required to make this prism in the middle-sized sand castle?

c.

What is the volume of sand required to make this prism in the largest sand castle?

d.

What is the perimeter of each of the squares that form the bases of each of the three different prisms in each of the three different sand castles?

e.

What is the total surface area of each of the rectangular prisms to be used in constructing each of the three sand castles? (This information is needed to construct nets for the molds that will be used to create the prisms.)

8.

Chau and Kassandra’s plans for the smallest sand castle include columns in the shape of cylinders, with the circular base having radii of . The height of the column is .

a.

What is the volume of sand required to make each of these columns in the smallest sand castle?

b.

What is the volume of sand required to make this column in the middle-sized sand castle?

c.

What is the volume of sand required to make this column in the largest sand castle?

d.

What is the circumference of each of the circles that form the circular bases of each of the three different columns in the three different sand castles?

e.

What is the total surface area of the cylinders—including the two circular bases and the rectangles that wrap around to form the cylinders—in each of the three sand castles? (This information is needed to construct the molds in which wet sand will be poured to create the columns.)

Ready for More?

We can visualize what happens to rectangular area when we double or triple the length and width of the sides, because we can see the smaller rectangle replicated in the scaled-up version of the rectangle. But how might we see this relationship with circles, since we can’t visualize the scaled-up circle being decomposed into four congruent smaller circles?

Using the following diagram, show that the four regions in the circle have the same area as one of the smaller circles.

Circle with two circles inside with diameters equivalent to the radius of the larger circle.

Takeaways

When the linear dimensions of a 3-D solid are multiplied by a scale factor of :

The surface area

The volume

This knowledge is useful because

Lesson Summary

In this lesson, we learned how area and volume scale if we scale the linear measures in a 3-D shape. Consequently, if we scale up a 3-D figure, we do not need to recalculate its surface area and volume using formulas for volume and area.

Retrieval

1.

Find the exact value of the missing side.

A right triangle with one side 16, hypotenuse 24, and one side ?

2.

Convert to radians.

3.

Convert to degrees.